# Levi-Civita Connection in Coordinates

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## Theorem

Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.

Let $\nabla$ is the Levi-Civita connection of $\struct {M, g}$.

Let $g$ be a Riemannian or pseudo-Riemannian metric, which locally reads:

- $g = g_{ij} \rd x^i \otimes \rd x^j$

where $\paren {g_{ij}}$ is a matrix of smooth functions.

Let $g^{ij}$ be the inverse of $\paren {g_{ij}}$.

Suppose $\Gamma^k_{ij}$ are the connection coefficients of $\nabla$.

Then in any smooth coordinate chart for $M$ we have:

- $\Gamma^k_{ij} = \dfrac 1 2 g^{kl} \paren {\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} }$

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## Proof

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## Source of Name

This entry was named for Tullio Levi-Civita.

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Symmetric Connections